The Physical Aspects of Fish Locomotion

THE PHYSICAL ASPECTS OF FISH LOCOMOTION

BY E. G. RICHARDSON, B.A., PH.D., D.SC. (Armstrong College, Newcastle on Tyne.)

(Received January i, 1935.) (With Eight Text-figures.) I. INTRODUCTION. THE mechanism by which a fish is able to dart through a heavy medium like water with little apparent effort has long been a matter of interest to zoologists. Before the arrival of the cinematograph in its modern form, Marey (1884) obtained a series of " chronophotographic" records of fishes in movement to demonstrate the sinusoidal motion, while more recently Magnan (1930) and Gray (1933) have taken cinematograph pictures at the rate of 10-20 per sec, in which the fish is seen swimming behind a reticule, thus making detailed study of the mode and rate of pro- gression possible. Houssay (1912) and Breder (1926) have also made contributions to this problem. Their work will be noted later. With the general adoption of streamlined form for all bodies which have to move rapidly through air or water, the attention of workers in hydrodynamics has been focused on the same problem. Indeed, the extensive researches of Magnan were subventioned by the French Air Ministry. It is from this aspect that the author is interested in the subject. There appear to be two possible explanations to account for the rapidity of movement of a fish through the water, viz. (1) that in virtue of the mucus on its surface it may possess a resistance lower than that of the same shape when unlubricated, (2) that in virtue of its flexibility it cleaves the water with less resistance than if rigid. If either of these facts can be substantiated it is obviously of great importance to aircraft and ship design.

II. RESISTANCE MEASUREMENTS. Although Houssay made measurements of the resistance of wooden models of fish towed through the lake of the ficole Militaire at Paris, the only measurements on fishes were those of Magnan deduced from cinema photographs of the weighted bodies of fish freely falling in a water tank. The photographs are scarcely clear enough for the exact deduction of the rate of fall. Indeed Magnan decides that the space : time curves are exact parabolas, which as he rightly deduces, implies that the resistance is independent of the velocity. He does not, however, complete the argument which is that if the parabolic law is followed the velocity increases without limit, a condition only fulfilled on free fall in a vacuum. A body falling in a 64 E. G. RICHARDSON resisting medium must sooner or later reach a constant velocity, determined by the balance between gravity and resistance, which can then be determined as a function of the terminal velocity. The author decided to determine the rate of fall of fishes over various distances by a direct chronographic method. A tank 8 ft. high and 1 ft. 6 in. square was filled with water and equipped with starting and stopping devices for the object whose fall was to be investigated. The former device was arranged so that the fish could be started from a little below the surface, the latter was a tray which could be lowered by cords passing over pulleys to counterpoise weights. The fish was held in an electromagnetic grip by its tail. Breaking the electric circuit released the fish and simultaneously released the brake of a Wood's chronograph. When the nose of the fish hit the lower stage, which was a pivoted gauze network, it broke a contact, which stopped the chronograph again. As will be seen from the diagram of the apparatus (Fig. 1) this contact had to be arranged in a vessel above water, otherwise it did not break smartly, the water being a conductor of electricity. With this apparatus the time could be read to 0-002 sec, while the distances were measured by telescope to 1 mm. The freshly killed fish, deprived of all its fins except the caudal fin, was loaded by lead shot in the mouth. Trouble was experienced at distances of fall exceeding 2 ft. owing to the flexibility of the fish which often led it to describe a curved trajectory. Later a wire stiffener was pushed into the body to keep the fall vertical. At any given distance, times were repeated until a minimum had been established. After each fall the lower stage was raised to retrieve the fish. When a series of falls at various distances and weights had been made, the fish was carefully measured and a template made. Subsequently a wooden model was made to the template, loaded in the scooped-out interior by lead shot and provided with an aluminium tail. The time of fall of the model in the tank was then determined under three conditions of surface, i.e. (a) roughened, (b) smoothed and polished, (c) greased with oil. Writing the equation of motion of the falling body in the form ,

m dt=mg~R' where the first term represents the downward force on the body at the instant when its velocity is v, the second term is the weight of the body in the water, or buoyancy, and the third is the resistance. We will write the latter as a function of v in the form kvn, so that , m =m g + kvn. at n may have, in practice, values between o and 2; (a) if n = o, v = ( ) t, the \m ml acceleration is constant and we have the case postulated by Magnan; (b) if n = 1 (Stokes' law), fc the terminal velocity is m'glmk; (c) if n = 2 (Newton's law), v = c tanhgjc, where c is the terminal velocity =Vm'glmk. In case (c) the terminal velocity is of course less, The Physical Aspects of Fish Locomotion 65 ceteris paribus, and reached in a shorter time. On Fig. 2 are plotted the displacement : time curves for a mackerel and a herring loaded by different weights to show that a terminal velocity, indicated by the graph becoming a straight line, is in

To starting switch of chronograph

[*o stopping switch of chronograph

Fig. 1. Tank for time of fall of fishes and models. practically every case attained within the depth of the tank. Probably Magnan's failure to detect a terminal velocity is due to his not having carried the distances far enough. (Although he had at his disposal a tank 3 metres deep, the displacement: time curves on which he bases his n = o deduction seem not to extend beyond half a metre.) Table I gives the resulting data in the form of weight in water (W) and corresponding terminal velocity (V). These are subsequently plotted in the form JEB-XIIli 5 66 E. G. RICHARDSON log W: log V on Fig. 3. The slope of these curves gives the appropriate value of n at each value of V.

3 4 5 Time (sees.) Fig. 2. Rate of fall of fishes

Table I. Resistance measurements. Mackerel. Weight in air 1OC3 gm., in water 267 gm., volume 123 c.c. W gm. 267 40-5 480 650 Fcm./sec 6-65 n-o 125 13-5 Rough wooden model. Weight in air 151 gm., in water 267 gm., volume 130 c.c. W 267 41-5 48-2 65-0 V 83 100 i2-o 13-3 Wooden model smoothed and varnished. W 267 41 -5 45-0 V 8-3 12-0 12-5 Herring. Weight in air 864 gm., in water 13-8 gm., volume 72-6 c.c. W 13-8 230 31-2 V S'O 100 12-0 Rough wooden model. Weight in air 84-2 gm., in water 13-9 gm., volume 70-3 c.c. W 13-9 23-0 31-0 V 5-0 9-0 no Wooden model smoothed and varnished. W 13-9 188 23-0 31-3 V 5"25 8'0 IO-O I2'2 The general conclusion to be drawn from this table and the graphs is that the fish has a slightly less resistance than the rough wooden model, but that a carefully smoothed and polished wooden model can improve on this, particularly at the low speeds. The exponent n given by the slope of the curves of Fig. 3 varies from i-o to i-8 for the fishes used, just below the turbulence regions to which Newton's law applies. It is probable that the normal velocity range covered by a fish lies below the turbulent regime except when it is spurred to violent movement. The Physical Aspects of Fish Locomotion 67 The experiments with the smoothed model covered with a light oil previous to immersion to imitate the mucus on the body of the fish did not indicate any marked

• fish o rough model X varnished model

1-0

1-2

1-0, 0-6 0-8 1-0 1-2 1-4 log terminal V Fig. 3. Resistance of fishes.

Paraffin +2

I Interface between / / / liquid

-2

0 I 2 Velocity (cm./sec.) Fig. 4. Velocity gradients in two liquids in relative motion. improvement on the resistance of the model at speeds below the critical. Both theory and practice, however, concur in indicating that the presence of a viscous layer, i.e. more viscous than the main body of the fluid in which it is immersed, 5-2 68 E. G. RICHARDSON delays the onset of turbulence. It is in fact the shearing force in the "boundary layer " of fluid close to the solid surface of an immersed obstacle that gives rise to instability which in turn leads to turbulence in the wake. The viscosity of the mucus was determined in a cup-and-ball viscometer and found to be about five times that of water at ordinary temperatures. Below critical speeds the added viscous layer wall simply follow the Stokes' type of viscous flow and cannot lead to any improvement in resistance if the velocity relative to the surface falls from its mid-stream value to zero at the surface without a break. It has been suggested, however, that there may be sliding of two liquids one over the other at their common interface, although it is established that none occurs at a solid-liquid boundary. Some experiments which the author performed some time ago give the lie to this suggestion (Richardson, 1933). One liquid (paraffin) was made to pass over another (water) in a tank, and by means of a hot-wire anemometer the velocity gradient on each side of and across the interface was measured. Typical results are shown in Fig. 4 for different surface speeds of the paraffin. No jump in the velocity occurs at the interface, merely a change of slope consequent upon the difference of viscosity. Except at high speeds then we may dismiss the idea that a fish has a small resistance because of its oiled surface.

III. EXPERIMENTS ON LOCOMOTION. Having failed to detect any morphological peculiarity in the fish which should give it a lowered resistance, we turn to the more difficult question of whether there is something inherent in the sinusoidal motion which should produce this effect. Difficult because it is not possible to consider the resistance apart from the propulsion. Just as with a screw propeller, measurements of the resistance at zero rate of advance, i.e. when the screw is acting as a mill, are not sufficient data for deriving the efficiency of the screw, so the resistance of the flexing fish should not be measured only when it is being towed or drifting in the stream. As it is not easy to make measurements on a live fish, one has recourse to a model, held stationary while the medium flows past it. Before describing such experiments, one should say at the outset that there is known to aeronautics a means of reducing the resistance of models which is pertinent to the fish problem. I refer to the Katzmayr (1922) effect, viz. the reduction of resistance sometimes experienced by an aerofoil when, instead of meeting the wind at a fixed angle of incidence it is given a periodic see-saw motion through a small angle. Alternatively, the object may be held still, while the incident fluid is given a sinusoidal motion by means of oscillating guides, this being the form that experiments on the Katzmayr effect usually take. Theory shows that the reduction in the drag component will be most apparent when the object is such that the lift component (or cross-force) varies rapidly with angle of incidence, and may in some cases attain a negative value, so that the aerofoil actually propels itself against the stream, some of the energy used to oscillate it thus becoming available for its propulsion. Owing to the difficulty of seeing how to apply the Katzmayr effect to full-scale aeronautics it has remained of merely academic interest, and an exact theory of the The Physical Aspects of Fish Locomotion 69 effect is not available. It seems, however, to be an effect which cannot be disre- garded in reference to fish locomotion, although an actual fish differs from the aerofoil model in that it is not rigid, and cannot oscillate rigidly about an axis as in the aeronautical experiments just described. The conditions of the experiments are however approximately fulfilled in the locomotion of those fishes whose bodies are too stiff to show a true sinusoidal motion, but merely flap to and fro, such as the mullet, cited by Magnan. Further, such fishes have bodies of the requisite form to exhibit a large change of cross-force as the body flaps to and fro, or, what amounts to the same thing, a small change of the angle of incidence of the stream on a long slim body will probably provoke a large change in the direction of the total (resisting) force. To study the resistance of a completely flexible body it was again necessary to resort to a model, this time a flexible rubber flag or "lamina". The experiments in

s o\—L- \» fr-%

Fig. 5. Wind channel and balance for resistance of flexible laminae.

the water channel having shown that the free-fall method was scarcely suited to a pliant model, owing to the tendency towards a curved trajectory, it was decided to measure the resistance in a wind-tunnel. The axis of the latter had to be vertical in order that the lamina, supported on a horizontal arm could hang vertical. The resistance of the flag was then measured by the position of a counterpoise weight on the balance arm (W, Fig. 5). The oscillation of the flag stalk (S) (corresponding to the head movements of the fish) was effected by making the stalk a thin steel rod of streamline section and fixing near it a little electromagnet (M), clamped rigidly to the balance beam (B). A dashpot (D) prevented vibration being imparted to the balance beam. The electromagnet was fed by the interrupted current from a Muir- head vibrator, whose frequency could be fixed at any value between 20 and 50 vibrations per sec, thus vibrating the flag head at a known and constant frequency and amplitude. The latter depended on the strength of the current supplied. The rubber was also on occasion weighted at the lower end by an aluminium strip to alter its natural frequency of vibration. In order to measure the velocity of the waves along the strip, it was illuminated intermittently by a ray of light shining 70 E. G. RICHARDSON through the stroboscopic disc and window in the channel as shown on the left of Fig. 5. By suitable adjustment of the speed of rotation of the stroboscopic disc the waves on the flexible strip could be "held" stationary and the wave-length ob- served through the travelling telescope X seen on the right. This wave-length varied slightly with the channel wind-speed, as did also the amplitudes at the head (the average value of the latter was 3 mm.). Knowing the frequency and the wave- length, the velocity of the waves along the lamina could be calculated. These corre- spond roughly with the channel wind velocities at zero resistance. Before discussing the results let us consider the theory of this type of fish propulsion considered as a motor. Let v = velocity of fish relative to water, and w = velocity of waves along fish, relative to the fish. Let the mass of the water set in motion per second—this will depend mainly on the amplitude of the wave passing down the body and pushing the water back—be mw. Let F= propulsive force = change of momentum, m (w — v) per sec. Then the energy usefully employed = Fv = mwv (w — v) per sec, while the kinetic energy rejected in the discharge = \mw (w — v)*, so that the total energy given to the fluid = Jmai {(w — v)* + zv (w — v)}. The theoretical efficiency is then zv {w — v)j(w'i — vi) = zvj{w + v). The actual efficiency will be less than this by an amount depending on the energy wasted in causing rotation of the water. It will be noted that though increase of w raises the force of propulsion it lowers the efficiency. The fish which approaches nearest to mechanical perfection is one which produces a slow wave of large amplitude along its body thus making m a maximum. Eddy production is likely to be increased if the fish tries to impart a high velocity to the water through a fast-moving wave as in lashing its tail. The results obtained with the flexible lamina are shown in Figs. 6 and 7 in the form of resistance/speed curves. It will be noticed that at two wind velocities the lamina shows negative resistance when subjected to forced vibrations. At higher wind speeds the resistance becomes positive, as might be expected since the velocity of the waves passing over the surface of the lamina approximates to or exceeds the velocity of the wind stream. So long as the conditions under which the fish is moving approximate to those of the lamina in these experiments it is obvious that the resistance overcome by the fish cannot be assumed to be the same as that overcome by a rigid fish moving at the same speed. The ordinary flapping of a flag in the wind is, of course, the corresponding dynamo effect to the fish-motor. The latter advances through the fluid by imparting energy to it, whereas the former is stationary and receives energy from the wind, so that any slight instability of the surface of the flag is passed along it as a wave of increasing amplitude. We find another analogy in the edge-tones formed at the mouth of a flute organ pipe where the air debouching from the slit hits the lip of the pipe and in doing so oscillates from side to side in flag fashion. Indeed, cinematograph films made by the author (Richardson, 1930, 1931 a, b) of the motion of such a jet, rendered suitably opaque, bear a striking similarity of movement to that in Gray's films of fish swimming. The late Lord Rayleigh treated the instability of flags and jets mathematically, The Physical Aspects of Fish Locomotion 71 but his results are scarcely useful in the present discussion. In the case of the edge- tone, the jet (owing to its extreme sensibility to disturbance) is set into strong rotational motion. Much of this energy goes into forming vortices, which are the

Model 33 MI.long

^Unloaded (mean w — 5 cm |iec, i +4 ^~">L«ded (mean a • 15 cm /we.) tz——•

V 1 1 -4 20 60 100 Air Velocity v cm./sec.) Fig. 6

Modtl 17 on. long

mean tr — 40 cm j

(mean w — 20 cm /iec )

20 60 100 Air Velocity v (cm./sec.) Fig. 7. Figs. 6 and 7. Resistance of flexible laminae; continuous lines = free vibration, broken lines = forced vibration at 25 cycles/sec.

cause of the sound, and these vortices roll between the jet and the stationary fluid on each side with half the velocity of the jet (as when a coin is rolled between the table and a card above). Whenever, in fact, relative motion of two fluids or of a flexible solid and a fluid occurs, there will be wave motion along the boundary, and vice versa. Breder constructed a mechanical wave machine which fish-like propelled itself through water in accordance with this principle.

IV. THE JET PROPULSION OF THE OCTOPUS. Of the available aids to translation, the majority of marine creatures employ sinusoidal motion to a greater or less extent, unless great acceleration is required, when a single flick may be given to the tail, or for reversing, when the pectoral fins may be used as oars. E. G. RICHARDSON

Table II.

Propulsion lb. wt. Efficiency v ft./sec. u> ft./sec. Theoretical Actual Theoretical Actual

12-5 o 0-051 o-°33 o-oo 000 22 9 0-166 0-044 0-58 030 23 21 0016 o-oio o-93 o-54 The members of the octopus family occasionally employ jet propulsion. They use the reaction derived from expelling a jet of fluid to propel themselves violently in the opposite direction. The theory of the jet propeller is similar to that already

I 0 40 50 ucm./sec.

-1

-2

Fig. 8. Resistance of jet propellers, -o 15 cm. model; x x x 22 cm. model; 10=30 cm./sec. approx. given for the wave propeller, save that w now stands for the velocity of the jet relative to the model, and m is the mass ejected per second. Some experiments made by the writer (1931 b) some years ago on air-jet propellers with a view to their possible adoption on aircraft are germane to this type of fish propulsion. A model with a streamlined head of wood, neatly fitted to a brass tube, was mounted in a wind channel with the tube pointing downstream. Air was led into the interior of The Physical Aspects of Fish Locomotion 73 the tube through the hollow strut which connected the model to the usual force balance. After passing along the barrel, the air was ejected from the stern. The weight of the model was taken up by a counterpoise on the balance beam, the de- flection of which acted as a measure of the propulsive force, which could be compared with that indicated by the propeller theory given above. Using a hot-wire anemometer and a Pitot manometer, the integrated velocity and the pressure drop behind the model were calculated, and hence the work done in overcoming friction in the apparatus was computed. This in turn allowed the actual efficiency to be compared with the theoretical value. Results for various jet and wind speeds are shown in Table II and on Fig. 8. The propulsive forces realised were smaller than the theoretical values, but the trend of the results was in agreement with theory. When v = w, the theoretical efficiency is 1, but the propulsive force is o. This case is nearly realised in the third experiment in Table II. Larger jet speeds could not be attained with the compressor available. Losses in friction inside the model were considerable in the arrangement chosen. Fig. 8 shows how the actual propulsive force varies (for a constant jet pressure) with the channel wind speed, (a) for a model having a 6-in. tube, and (b) for a model with a 9-in. tube. The size of the orifice has no effect on the efficiency, for although a greater area of discharge will produce a greater reaction, yet more energy must be supplied to maintain the velocity of discharge. The experiments with the model jet propellers verify in a general way the propeller theory as it applies to creatures or machines which use reaction for locomotion.

V. SUMMARY AND ACKNOWLEDGMENTS. 1. The resistance of an inert fish moving in water is approximately the same as that of a wooden model similar in form and moving at the same velocity. 2. When a lamina is made to vibrate in a current of air, it exerts a thrust in the direction in which the vibrations are travelling over the lamina; as long as the velocity of movement of the vibrations exceeds that of the external current of air, the resistance of the lamina is negative. 3. Attention is drawn to the significance of the Katzmayr effect in respect to the locomotion of fish.

I wish to express my best thanks to Dr H. O. Bull of the Dove Marine Laboratory for his help during the course of this work, and for his criticism of the manuscript. I am also indebted to Dr J. Gray for helpful criticism and to Dr R. Curry and Mr W. Ridley for their help in making the observations. 74 E. G. RICHARDSON

REFERENCES. BREDER, C. M. (1026). Zoologica, N.Y., 4, 159. GRAY, J. (1933). J. exp. Biol. 10, 88, 391; Proc. roy. Soc. 113, 115. HOUSSAY, F. (1912). Forme, Puistance et Stability des Poissotu. KATZMAYR, Z. (1922). Flugtech. Bibl. 13, 65, 80. MAGNAN, A. (1930). Arm. Set. nat. 13, 355. MAREY, F. (1884). Le Moiwement. RICHARDSON, E. G. (1930). Proc. roy. Soc. Arts, Jan. (1931 a). Proc. phys. Soc. Lond. 43, 394. (1931 b). J.R. aero. Soc. Jan. (i933)- Kolloidzschr. 65, 32. Other references to Fish Locomotion in Russell, E. S. and Bull, H. O. (1933), J. Cons. int. Explor. Mer. 7, 255.